# Families of Segre fourfolds with a view to del Pezzo fibrations

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Marcello Bernardara

created by risa on 09 Apr 2018

12 apr 2018
-- 14:30

Aula 211, Dip.Matematica, Università "Roma Tre", Roma

**Abstract.**

Fibrations in del Pezzo surfaces of degree 6 are an interesting case
of Mori fiber spaces: for example, special cubic fourfold of
discriminant 18 admit such a fibration and their rationality is
related to it, as shown by
Addington-Hassett-Tschinkel-Varilly-Alvarado. Recently, Kuznetsov
described a semiorthogonal decomposition for such fibrations.
In this talk, I will present a general construction of a Segre
fourfold fibration X-->M with simple degenerations. Namely, a flat map
X-->M whose general fiber is isomorphic to P^{2} imes P^{2} with a
natural embedding in a P^{8}-bundle over M. Such an X is described by a
double cover S o M ramified along the degeneracy locus and an Azumaya
algebra B of order 3 over S, and comes with a natural Lefschetz
decomposition with respect to map into the P^{8} bundle. Such a
fibration comes with a natural dual fibration Z-->M in determinantal
cubic hypersurfaces of P^{8} and a (categorical) resolution of such.
As an application of this construction, we aim to give a recipe to
construct del Pezzo fibrations of degree 6 over M as double linear
sections of such an X, and reconstruct Kuznetsov's semiorthogonal
decomposition via relative homological projective duality, as well as
fibrations in cubic surfaces over M with determinantal generic fiber.
This is a work in progress with Addington, Auel and Faenzi.